which is a continuous time binary branching process with birth rate \(\beta \)  and death rate \(\gamma \) . This process goes extinct before time \(t\)  with probability (see the formula in the middle of page 108 in [1]})
\( \left(\frac{\gamma e^{N(\beta -\gamma )t}-N^{-1}}{\beta e^{N(\beta -\gamma )t}-\gamma }\right)^{Nx_1}.\) 
